Matrix inverse subspaces

Now we are ready to formulate the basic idea of the correction algorithm in order to correct the four-indexed operator f1, it is enough to correct the two-indexed operators fc and f 1 in the supermatrix representation (7.100). The real advantage of this proposal is its compatibility with any definite way of f2 and P7 correction [61, 294], The matrix inversion demanded in (7.99) is divided into two stages. In the fi, v subspace it is possible to find the inverse matrix analytically with the help of the Frobenius formula that is well known in matrix algebra [295]. The... 

C) Subspaces, Linear (In)dependence, Matrix Inverse and Bases... 

Usually, p is chosen to be a number between 4 and 10. In this way the system moves in the best direction in a restricted subspace. For this subspace the second-derivative matrix is constructed by finite differences from the stored displacement and first-derivative vectors and the new positions are determined as in the Newton-Raphson method. This method is quite efficient in terms of the required computer time, and the matrix inversion is a very small fraction of the entire calculation. The adopted basis Newton-Raphson method is a combination of the best aspects of the first derivative methods, in terms of speed and storage requirements, and the more costly full Newton-Raphson technique, in terms of introducing the most important second-de-... 

In order for Am to be a regular matrix at every point in the assumed region of configuration space it has to have an inverse and its elements have to be analytic functions in this region. In what follows, we prove that if the elements of the components of Xm are analytic functions in this region and have derivatives to any order and if the P subspace is decoupled from the corresponding Q subspace then, indeed. Am will have the above two features. 

The preceding definition of a kinetic SDE reduces to that given by Hiitter and Ottinger [34] in the case of an invertible mobility matrix X P, for which Eq. (2.268) reduces to the requirement that Zap = K. In the case of a singular mobility, the present definition requires that the projection of Z p onto the nonnull subspace of K (corresponding to the soft subspace of a constrained system) equal the inverse of within this subspace, while leaving the components of Z p outside this subspace unspecified. 

We next present a relationship between the inverses of the projected tensors S and T within the soft and hard subspaces, respectively. By requiring that the matrix product of the block matrices (A.l) and (A.2) yield the identity tensor, it is straightforward to show that the elements of the matrix... 

Note that we require the inverse of the matrix (1 - M) within the 0 subspace. Now multiplying both sides of Eq. (7.6) by P, substituting QG(a))P from Eq. (7.7), and again multiplying by P on both sides yields... 

The calculation of cannot be carried out analytically54 unless we use a truncated form for W(q). However, when W embodies only short-range interactions, the calculation (2.74) amounts to the inversion of a small matrix.56 In the case of (2.60), where the intermolecular interactions are purely electronic, this approximation amounts to limiting the range of J to the near neighbors. For a domain Q> of action of W around a given site, W acts in the finite subspace... 

The effective Fock matrix (20) is in our implementation [51] the quantity which is averaged in the optimization based on the direct inversion in the iterative subspace (DIIS) method [52],... 

We will now derive a Dyson equation by expressing the inverse matrix of the extended two-particle Green s function Qr,y, u ) by a matrix representation of the extended operator H. We already mentioned that the primary set of states l rs) spans a subspace (the model spaice) of the Hilbert space Y. Since the states IVrs) are /r-orthonormal they are also linearly independent and thus form a basis of this subspace. Here and in the following the set of pairs of singleparticle indices (r, s) has to be restricted to r > s for the pp and hh cases (b) and (c) where the states are antisymmetric under permutation of r and s. No restriction applies in the ph case (a). The primary set of states Yr ) can now be extended to a complete basis Qj D Yr ) of the Hilbert space Y. We may further demand that the states Qj) are /r-orthonormal ... 

The size of the f a) subspace is quite large for reasonable basis sets. For N electrons and a spin orbital basis of rank K, there are K — N) K — 2)N/2 elements in the orthogonal complement space. The solution of the Dyson-like equation would require the inversion of a matrix of this dimension for every value of the energy parameter. This approximation of the self-energy first introduced by J. Linderberg and Y. Ohrn,, has been discussed by G. D. Purvis III, and Y. Ohrn, and J. Schirmer and L. S. Cederbaum . A simplified version of this approach is the so-called Diagonal 2p-h TDA , which becomes... 

In iteration n the A matrix has dimension (n -1- 1) x (n -t 1), where n usually is less than 20. The coefficients c can be obtained by directly inverting the A matrix and multiplying it onto the b vector, i.e. in the subspace of the iterations the linear equations are solved by direct inversion , thus the name DIIS. Having obtained the coefficients that minimize the error function at iteration n, the same set of coefficients is used for generating an extrapolated Fock matrix (F ) at iteration n, which is used in place of F for generating the new density matrix. 

The matrix A transforms the vector space X, defined by the vector X, into a subspace of the vector space B, defined by the vector b. In order to solve the equation Ax = b, we seek the solution vector x. Using the definition of the inverse of A, A A — J, we have... 

Rather than straightforward application of O Eq. 5.5, the response equations ( Eq. 5.4) are solved in an iterative procedure in which the matrices and vectors are projected onto a small subspace. In this procedure, an initial guess (a trial vector) is generated according to O Eq. 5.5 by neglecting all off-diagonal elements of H and S such that the inverse matrix [T-L -is readily computed. The iterations proceed by refining the trial vector until the norm of the residual vector... 

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